Homological algebra of pro-Lie Polish abelian groups

TL;DR

This paper develops homological algebra for pro-Lie Polish abelian groups, characterizing their injective and projective objects, and analyzing their homological dimensions within various categories.

Contribution

It extends the homological algebra framework to pro-Lie Polish abelian groups, providing new characterizations and structural insights.

Findings

Pro-Lie Polish abelian groups form a thick subcategory of Polish abelian groups.

The category has enough projectives but not enough injectives, with homological dimension 1.

Categories of non-Archimedean Polish abelian groups and certain torsion groups have enough injectives and projective objects.

Abstract

In this paper, we initiate the study of pro-Lie Polish abelian groups from the perspective of homological algebra. We extend to this context the type-decomposition of locally compact Polish abelian groups of Hoffmann and Spitzweck, and prove that the category $\mathbf{proLiePAb}$ of pro-Lie Polish abelian groups is a thick subcategory of the category of Polish abelian groups. We completely characterize injective and projective objects in $\mathbf{proLiePAb}$. We conclude that $\mathbf{proLiePAb}$ has enough projectives but not enough injectives and homological dimension $1$. We also completely characterize injective and projective objects in the category of non-Archimedean Polish abelian groups, concluding that it has enough injectives and projectives and homological dimension $1$. Injective objects are also characterized for the categories of topological torsion Polish abelian groups and for Polish abelian topological $p$-groups, showing that these categories have enough injectives and homological dimension $1$.